\(\int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx\) [558]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {3 \sqrt {5-2 x} \sqrt {2+3 x+x^2}}{2 (4+3 x)}-\frac {3 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{2 \sqrt {2+3 x+x^2}}+\frac {23 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{18 \sqrt {2+3 x+x^2}}-\frac {19 \sqrt {-2-3 x-x^2} \operatorname {EllipticPi}\left (\frac {3}{2},\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{36 \sqrt {2+3 x+x^2}} \] Output:

3*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)/(8+6*x)-3/2*(-x^2-3*x-2)^(1/2)*EllipticE 
((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)+23/18*(-x^2-3*x-2)^(1/2)*Ellip 
ticF((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)-19/36*(-x^2-3*x-2)^(1/2)*E 
llipticPi((2+x)^(1/2),3/2,1/3*2^(1/2))/(x^2+3*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 20.75 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {\frac {621 \sqrt {5-2 x} \left (2+3 x+x^2\right )}{4+3 x}-621 \sqrt {1+x} \sqrt {2+x} E\left (\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )|\frac {7}{9}\right )+92 \sqrt {1+x} \sqrt {2+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right ),\frac {7}{9}\right )-38 \sqrt {1+x} \sqrt {2+x} \operatorname {EllipticPi}\left (\frac {21}{23},\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right ),\frac {7}{9}\right )}{414 \sqrt {2+3 x+x^2}} \] Input:

Integrate[Sqrt[5 - 2*x]/((4 + 3*x)^2*Sqrt[2 + 3*x + x^2]),x]
 

Output:

((621*Sqrt[5 - 2*x]*(2 + 3*x + x^2))/(4 + 3*x) - 621*Sqrt[1 + x]*Sqrt[2 + 
x]*EllipticE[ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9] + 92*Sqrt[1 + x]*Sqrt[2 + 
 x]*EllipticF[ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9] - 38*Sqrt[1 + x]*Sqrt[2 
+ x]*EllipticPi[21/23, ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9])/(414*Sqrt[2 + 
3*x + x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.97 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1285, 2154, 1269, 1172, 27, 321, 327, 1279, 27, 186, 27, 412}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {5-2 x}}{(3 x+4)^2 \sqrt {x^2+3 x+2}} \, dx\)

\(\Big \downarrow \) 1285

\(\displaystyle \frac {1}{4} \int \frac {6 x^2+16 x+17}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 2154

\(\displaystyle \frac {1}{4} \left (\int \frac {2 x+\frac {8}{3}}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx+\frac {19}{3} \int \frac {1}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{4} \left (\frac {23}{3} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx+\frac {19}{3} \int \frac {1}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{4} \left (\frac {19}{3} \int \frac {1}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {46 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {19}{3} \int \frac {1}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {46 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{3 \sqrt {x^2+3 x+2}}-\frac {2 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{4} \left (\frac {19}{3} \int \frac {1}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {2 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}+\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{4} \left (\frac {19}{3} \int \frac {1}{\sqrt {5-2 x} (3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 1279

\(\displaystyle \frac {1}{4} \left (\frac {38 \sqrt {x+1} \sqrt {x+2} \int \frac {1}{2 \sqrt {5-2 x} \sqrt {x+1} \sqrt {x+2} (3 x+4)}dx}{3 \sqrt {x^2+3 x+2}}+\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {19 \sqrt {x+1} \sqrt {x+2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x+1} \sqrt {x+2} (3 x+4)}dx}{3 \sqrt {x^2+3 x+2}}+\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {1}{4} \left (-\frac {38 \sqrt {x+1} \sqrt {x+2} \int \frac {2}{(23-3 (5-2 x)) \sqrt {2 x+2} \sqrt {2 x+4}}d\sqrt {5-2 x}}{3 \sqrt {x^2+3 x+2}}+\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (-\frac {76 \sqrt {x+1} \sqrt {x+2} \int \frac {1}{(23-3 (5-2 x)) \sqrt {2 x+2} \sqrt {2 x+4}}d\sqrt {5-2 x}}{3 \sqrt {x^2+3 x+2}}+\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {1}{4} \left (\frac {46 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{9 \sqrt {x^2+3 x+2}}-\frac {6 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {76 \sqrt {x+1} \sqrt {x+2} \operatorname {EllipticPi}\left (\frac {27}{23},\arcsin \left (\frac {1}{3} \sqrt {5-2 x}\right ),\frac {9}{7}\right )}{69 \sqrt {7} \sqrt {x^2+3 x+2}}\right )+\frac {3 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}{2 (3 x+4)}\)

Input:

Int[Sqrt[5 - 2*x]/((4 + 3*x)^2*Sqrt[2 + 3*x + x^2]),x]
 

Output:

(3*Sqrt[5 - 2*x]*Sqrt[2 + 3*x + x^2])/(2*(4 + 3*x)) + ((-6*Sqrt[-2 - 3*x - 
 x^2]*EllipticE[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] + (46*Sqrt[ 
-2 - 3*x - x^2]*EllipticF[ArcSin[Sqrt[2 + x]], 2/9])/(9*Sqrt[2 + 3*x + x^2 
]) - (76*Sqrt[1 + x]*Sqrt[2 + x]*EllipticPi[27/23, ArcSin[Sqrt[5 - 2*x]/3] 
, 9/7])/(69*Sqrt[7]*Sqrt[2 + 3*x + x^2]))/4
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

rule 1285

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.) 
*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*Sqrt[f + g*x]* 
(Sqrt[a + b*x + c*x^2]/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*( 
m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqr 
t[a + b*x + c*x^2]))*Simp[2*c*d*f*(m + 1) - e*(a*g + b*f*(2*m + 3)) - 2*(b* 
e*g*(2 + m) - c*(d*g*(m + 1) - e*f*(m + 2)))*x - c*e*g*(2*m + 5)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x 
] && LtQ[m, 0] &&  !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {2 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{21 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {\sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{14 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {3 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{2 \left (3 x +4\right )}-\frac {19 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right )}{966 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(249\)
risch	\(-\frac {3 \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{2 \left (3 x +4\right ) \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}-\frac {\left (\frac {2 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{63 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {\sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{42 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {19 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticPi}\left (\frac {\sqrt {35-14 x}}{7}, \frac {21}{23}, \frac {\sqrt {7}}{3}\right )}{2898 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(288\)
default	\(\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (138 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) x +1449 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) x +114 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right ) x +184 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+1932 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+152 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right )+5796 x^{3}+2898 x^{2}-31878 x -28980\right )}{1932 \left (2 x^{3}+x^{2}-11 x -10\right ) \left (3 x +4\right )}\)	\(290\)

Input:

int((5-2*x)^(1/2)/(3*x+4)^2/(x^2+3*x+2)^(1/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

(-(-5+2*x)*(x^2+3*x+2))^(1/2)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2)*(-2/21*(5-2* 
x)^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)/(-2*x^3-x^2+11*x+10)^(1/2)*Elliptic 
F(1/3*(5-2*x)^(1/2),3/7*7^(1/2))-1/14*(5-2*x)^(1/2)*(14*x+14)^(1/2)*(2*x+4 
)^(1/2)/(-2*x^3-x^2+11*x+10)^(1/2)*(7/2*EllipticE(1/3*(5-2*x)^(1/2),3/7*7^ 
(1/2))-EllipticF(1/3*(5-2*x)^(1/2),3/7*7^(1/2)))+3/2/(3*x+4)*(-2*x^3-x^2+1 
1*x+10)^(1/2)-19/966*(5-2*x)^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)/(-2*x^3-x 
^2+11*x+10)^(1/2)*EllipticPi(1/3*(5-2*x)^(1/2),27/23,3/7*7^(1/2)))
 

Fricas [F]

\[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int { \frac {\sqrt {-2 \, x + 5}}{\sqrt {x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )}^{2}} \,d x } \] Input:

integrate((5-2*x)^(1/2)/(4+3*x)^2/(x^2+3*x+2)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5)/(9*x^4 + 51*x^3 + 106*x^2 + 96 
*x + 32), x)
 

Sympy [F]

\[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int \frac {\sqrt {5 - 2 x}}{\sqrt {\left (x + 1\right ) \left (x + 2\right )} \left (3 x + 4\right )^{2}}\, dx \] Input:

integrate((5-2*x)**(1/2)/(4+3*x)**2/(x**2+3*x+2)**(1/2),x)
 

Output:

Integral(sqrt(5 - 2*x)/(sqrt((x + 1)*(x + 2))*(3*x + 4)**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int { \frac {\sqrt {-2 \, x + 5}}{\sqrt {x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )}^{2}} \,d x } \] Input:

integrate((5-2*x)^(1/2)/(4+3*x)^2/(x^2+3*x+2)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-2*x + 5)/(sqrt(x^2 + 3*x + 2)*(3*x + 4)^2), x)
 

Giac [F]

\[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int { \frac {\sqrt {-2 \, x + 5}}{\sqrt {x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )}^{2}} \,d x } \] Input:

integrate((5-2*x)^(1/2)/(4+3*x)^2/(x^2+3*x+2)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-2*x + 5)/(sqrt(x^2 + 3*x + 2)*(3*x + 4)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int \frac {\sqrt {5-2\,x}}{{\left (3\,x+4\right )}^2\,\sqrt {x^2+3\,x+2}} \,d x \] Input:

int((5 - 2*x)^(1/2)/((3*x + 4)^2*(3*x + x^2 + 2)^(1/2)),x)
 

Output:

int((5 - 2*x)^(1/2)/((3*x + 4)^2*(3*x + x^2 + 2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {5-2 x}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{9 x^{4}+51 x^{3}+106 x^{2}+96 x +32}d x \] Input:

int((5-2*x)^(1/2)/(4+3*x)^2/(x^2+3*x+2)^(1/2),x)
 

Output:

int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(9*x**4 + 51*x**3 + 106*x**2 + 
 96*x + 32),x)